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\hyphenation{Zeit-schrift De-de-kind Ra-don pre-print Knas-ter}

\loadeusm
\loadeurm

\Centerline{\bf Problems}

\Centerline{\smc D.H.Fremlin}

\Centerline{\it University of Essex, Colchester, England}

\medskip

{\it In an attempt to add to the gaiety of nations, I am offering
cash prizes for most of the questions below.   The prizes, indicated
by the formulae `(\$n)' following questions, are not for solving the
problems but for communicating the solutions (whether yours or
others') to me.   They will be paid in cheques drawn on a UK bank.
I may pay out part of a prize for a partial solution or a consistency
result if it seems to me to be a substantial step forward.

\hfill David Fremlin \hskip3em
}

\medskip

\noindent {\bf Notation} Most terms may be found in {\smc Fremlin 74}
or {\smc Fremlin 84};  see also {\smc Fremlin 02} and {\smc Fremlin 03}.
$\frak{m}$ is the least cardinal for which
MA($\frak{m}$) is false;  thus $`\frak{m}=\frak{c}$' is Martin's
axiom.
$\frak{p}=\frak{m}_{\sigma\text{-centered}}$ is the least cardinal
for which MA$_{\sigma\text{-centered}}(\frak{p})$ is false.
$\frak{b}$ is the least cardinal of any subset $A$ of $\Bbb N^{\Bbb N}$
such that there is no $g \in \Bbb N^{\Bbb N}$ for which
$\{ n:f(n) > g(n) \}$ is finite for every $f \in A$.

For problems marked $\dagger$ you can find notes in {\tt
https://www1.essex.ac.uk/maths/staff{\vs}fremlin{\vs}problems{\penalty-100}.htm}.

\medskip

{\bf D.} Let $(X,\Sigma,\mu)$ be a measure space.   A {\bf lifting} of
$\mu$ is a Boolean homomorphism $\theta:\Sigma\to\Sigma$ such that
$\theta E=\emptyset$ whenever
$\mu E=0$ and $\mu(E\triangle\theta E)=0$ for every
$E\in\Sigma$.   It is known that (a) every complete probability space
has a lifting (see {\smc Fremlin 02}, \S341) (b) subject
to the continuum hypothesis, every probability space of Maharam
type at most $\frak{c}^+$ has a lifting (see
{\smc Fremlin 08}, \S535) (c) it is possible (that
is, relatively consistent with ZFC) that the restriction of Lebesgue
measure to the Borel
$\sigma$-algebra of $[0,1]$ has no lifting (see {\smc Shelah 83} or
{\smc Just 92} or {\smc Burke 93}).

{\bf Problem} Must there be (that is, is it a theorem of ZFC) that there
is a probability space without a lifting?  (\pound33.)

{\bf Remark} By Maharam's theorem, it is enough to consider
$\nu_{\kappa}\restr\Sigma$ where $\kappa$ is a
cardinal, $\Sigma$ is the Baire $\sigma$-algebra of $\{0,1\}^{\kappa}$
and $\nu_{\kappa}$ is the usual measure on $\{0,1\}^{\kappa}$.   (The
{\bf Baire $\sigma$-algebra} of a topological space is the smallest
$\sigma$-algebra with respect to which every continuous function is
measurable;  for a compact zero-dimensional space, as in this case, it
is the $\sigma$-algebra generated by the open-and-closed sets.)
What about $\kappa=\omega_3$?      (\pound25.)

See also {\bf DO, EJ} below.

\medskip

{\bf F.} Let $K$ be a compact convex set in a (non-locally-convex)
Hausdorff linear topological space $E$, and $U$ a neighbourhood
of $0$ in $E$.   Is there necessarily a continuous function
$T:E\rightarrow K$ such that $Tx - x\in U$ for every
$x\in K$?   What if $E=L^0([0,1])$?   (\pound2.)

\medskip

{\bf H.} Let $E$ be a Banach lattice which is a symmetric sequence
space
(i.e., a solid linear subspace of $\BbbR^{\Bbb N}$ which is invariant
under permutations of $\Bbb N$).   Must $E$ have the approximation
property?

\medskip

{\bf R.} (W.Moran) Can it be that

\Centerline{$\lim_{n\rightarrow\Cal F}E_n
=\{\alpha:\{ n :\alpha\in E_n\}\in\Cal F\}$}

\noindent is negligible whenever
$\langle E_n\rangle_{n\in\Bbb N}$ is a stochastically independent
sequence of measurable subsets of $[0,1]$ such that
$\lim_{n\to\infty}\mu E_n=0$, and $\Cal F$ is a non-principal
ultrafilter on $\Bbb N$?
(\pound40.)   (If $\frak{m}=\frak{c}$, no;  see {\smc Fremlin
\& Talagrand 79} and
{\smc Talagrand 84}, 9-1-3.   Also {\smc Shelah \& Fremlin 93}.)

\medskip

{\bf W$\dagger$.} (L.Schwartz) A {\bf Borel measure} on a topological
space $X$
is a measure with domain the Borel $\sigma$-algebra of $X$.   A
Hausdorff topological space $X$ is
{\bf Radon} if every totally finite Borel measure on X is inner regular
with respect to
the compact sets.   Can it be that the product of
two Radon spaces is always Radon?  (\pound30.)   (If {\it either}
there is an atomlessly-{\hskip0.001em}measurable cardinal {\it or}
$\frak{m}=\frak{c}$, no;  see {\smc Wage 80} and {\smc Fremlin n82a}.)

See also {\bf DZ}, {\bf FM} below.

\medskip

{\bf X.} Let $E$ be a Dedekind complete Banach lattice for which the
norm is
Fatou and the norm topology is Levi (i.e., if $A$ is a non-empty
upwards-directed subset of the unit ball of $E$, then sup\/$A$ exists
in $E$ and $\|\sup A\|\le 1$).   Does it follow that there is a
positive contractive
projection from the bidual $E^{**}$ of $E$ onto $E$?  (\pound8.)
(See {\smc Schaefer 74}, \S IV.4.)

\medskip

{\bf Z$\dagger$.} Assume that $\frak m>\omega_1$.
Give $\BbbR^{[0,1]}$ the topology $\frak{T}_p$ of pointwise
convergence, and write $\eusm L^0(\Cal B)$ for the subset of
$\BbbR^{[0,1]}$ consisting of Borel measurable functions.   For
$K\subseteq\BbbR^{[0,1]}$ write $\overline{\Gamma(K)}$ for its
$\frak{T}_p$-closed convex hull in $\BbbR^{[0,1]}$.   Is
$\overline{\Gamma(K)}$ a subset of $\eusm L^0(\Cal B)$ whenever
$K\subseteq\eusm L^0(\Cal B)$ is
$\frak{T}_p$-compact?   (\pound8.)  (If $\frak{c}=\omega_1$,
no;  see {\smc Talagrand 84}, 10-1-1, or {\smc Fremlin 03}, 463Ye.  If
there is a real-valued-measurable cardinal, yes;  see 
{\smc Fremlin n95}.)

\medskip

{\bf AB$\dagger$.} 
If $E$ is a Riesz space, a norm $\|\thinspace\|$ on $E$ is
{\bf weakly Fatou} if it is a Riesz norm (that is, $\|x\|\le\|y\|$
whenever $|x|\le|y|$) and there is a constant
$\alpha\ge 0$ such that $\|x\|\le\alpha\,\text{sup}_{y\in A}\|y\|$ 
whenever $A$ is a non-empty upwards-directed subset of $E$
with supremum $x$.   Many of the results proved in {\smc Fremlin 74}
for Fatou norms are true also for weakly Fatou norms.   There are Banach
lattices with weakly Fatou norms which have no equivalent Fatou norms
({\smc Elliott p18}).   But is there a weakly Fatou norm on $C([0,1])$
which is not equivalent to any Fatou norm?  (\pound9.)

\medskip

{\bf AC$\dagger$.} (A.Bellow)  Can there be a probability
space $X$ and a set $K$ of real-valued measurable functions on $X$ such
that (i) $K$ is compact for the topology $\frak{T}_p$ of pointwise
convergence inherited from $\BbbR^X$ (ii) $K$ is separated by the
measure, that is, if two functions in $K$ agree almost everywhere they
are identical (iii) $K$ is not metrizable for $\frak{T}_p$?
(\pound40.)   (If $\frak{m}=\frak{c}$, no;  see {\smc Fremlin 08}, \S536.)
What if $X$ is a completely
regular Hausdorff space, $K$ consists only of continuous functions, and
$\mu$ is a Borel measure which is {\bf $\tau$-additive}, that is,
$\mu(\bigcup\Cal G)=\sup_{G\in\Cal G}\mu G$ whenever $\Cal G$ is an
upwards-directed family of open sets?   (\pound14.)

\medskip

{\bf AD.} (J.Bourgain \& F.Delbaen.)  Give $\BbbR^{[0,1]}$ the
topology $\frak{T}_p$ of pointwise convergence, and write
$\eusm L^0(\Sigma)$ for the subset of $\BbbR^{[0,1]}$ consisting of
Lebesgue measurable
functions.   For $A\subseteq\BbbR^{[0,1]}$ write $\overline{A}$
for its $\frak{T}_p$-closure in $\BbbR^{[0,1]}$.   Can there be a
countable set $A$ such that (i)
$\overline{A}$ is $\frak{T}_p$-compact (ii)
$\overline{A}\subseteq\eusm L^0(\Sigma)$  (iii) there is a function in
$\overline{A}$ which is
not the limit almost everywhere of a sequence in $A$?  (\pound21.)
(If $\frak{m}=\frak{c}$, no;  see {\smc Talagrand 84}, 9-5-2.   If
$\overline{A}$ is countably tight, no;  see {\smc Fremlin 03}, 463E.)
What if $\overline{A}$ is convex?  (\pound19.)
(See {\smc Shelah \& Fremlin 93}.)

An equivalent form of the problem (if $\overline{A}$ is not assumed
to be convex) is:  Can there be a sequence
$\langle E_n\rangle_{n\in\Bbb N}$ of measurable subsets of $[0,1]$ such
that (i) $\lim_{n\to\Cal F}E_n$ is measurable for every ultrafilter
$\Cal F$ on $\Bbb N$ (see {\bf R} above) (ii) every finite subset of
$[0,1]$ is included in some $E_n$ (iii) $\lim_{n\to\infty}\mu_LE_n=0$?

\medskip

{\bf AM.} (Sierpi\'nski.)  Is there a Borel subset of the plane which
meets every straight line in exactly two points?   (\pound34.)   (See
{\smc Mauldin 98}.)

\medskip

{\bf AT.} [Efimov] Must there be an infinite compact
Hausdorff space $K$ such that neither $\beta\Bbb N$ nor $\omega+1$ (with
its order topology) can be embedded homeomorphically as a subset of $K$?
(\pound23.)
(If $\cf[\frak s]^{\le\omega}=\frak s$ and $2^{\frak s}<2^{\frak c}$, yes;  
see {\smc Dow 05}.   If $\frak b=\frak c$, yes;  see
{\smc Dow \& Shelah 13}.   In random real models, yes;  see 
{\smc Dow \& Fremlin 07}.)   

\medskip

{\bf BC.}  Let $F\subseteq\Bbb R$ be an F$_{\sigma}$ set and
$G\subseteq\Bbb R$ a G$_{\delta}$ set.   Write $\Cal I$ for
the $\sigma$-ideal of subsets of $E=F\cup G$ generated by
the compact subsets of $E$.   Is $\Cal I$ necessarily
$\frak{m}$-additive? (\pound9.)
(See {\smc Engelen Kunen \& Miller 89}.)

\medskip


{\bf BH.} A Radon measure $\mu$ on a compact Hausdorff space $X$ is
{\bf uniformly regular} if there are a compact metric space
$Z$ and a continuous function $f:X\rightarrow Z$ such that
$\mu f^{-1}[f[F]]=\mu F$ for every closed $F\subseteq X$.
Assuming that $\frak m>\omega_1$, does every Radon measure
on a first-countable compact Hausdorff space have to be uniformly
regular?  (\pound32.)   (See {\smc Fremlin 84}, 32L-N,
{\smc Plebanek 00} and {\smc Fremlin 08}, \S533.)

\medskip

{\bf BQ.} Assume that $\frak{m}=\frak{c}=\omega_2$.

{\bf (a)} Must there be a metric space
$X$ such that every separable subset of $X$ is Borel, but not
every subset of $X$ is Borel?   (If $\diamondsuit_{\frakc}(E)$
is true ({\smc Daniels \& Gruenhage 87}, B2Hb), yes.) (\pound2.)

{\bf (b)} Can it be that whenever
$X$ is a metric space with Borel sets of every class
$<\omega_1$, then $X$ has a separable subset of cardinal
$\frak{c}$? (\pound1).   (If $\diamondsuit_{\frakc}(E)$ is true or
$\square_{\omega_{\omega}}$ is true, no.)

{\bf (c)} If $X$ is a metric space in which every subset is Borel, does
every subset have to be F$_{\sigma}$?  (\pound9.)   (See
{\smc Fremlin 84}, \S23, and {\smc Fremlin Hansell \& Junnila
83}.)

\medskip

{\bf BU.} Assume that $\frak{m}>\omega_1$.   Can there be an
$\alpha<\omega_1$ such that
$\omega_1\nrightarrow (\omega_1,\alpha)^2$?   (\pound12.)
(If {\it either} $\alpha$ is less than or equal to the ordinal power
$\omega^2$ {\it or} PFA is true, no;  see {\smc Fremlin n99},
S42A, and {\smc Fremlin n86a}, 6E.)

\medskip

{\bf BY.}  Let K be the statement `any upwards-ccc partially ordered
set satisfies Knaster's condition upwards' (see {\smc Fremlin 84}, 41Ab,
and {\smc Todor\v cevi\'c 89}, \S7).  Does K imply that
$\frak{m}>\omega_1$?   (\pound26.)   Does K imply that
$2^{\omega_1}=\frak c$?   (\pound9.)

\medskip

{\bf CA.} (A.H.Stone.) Can there be metric
spaces $X$ and $Y$ and a Borel measurable function $f:X\rightarrow Y$
such that the Borel classes of $f^{-1}[H]$, as $H$ runs over the open
subsets of $Y$, are unbounded in $\omega_1$?  (\pound30.)  (Under any of a
large number of special axioms, no;  see {\smc
Fremlin Hansell \& Junnila 83}, Theorem 16.)

\medskip

{\bf CC$\dagger$.} If $X$ and $Y$ are topological spaces, a relation
$R\subseteq X\times Y$ is {\bf lower Borel measurable} if $R^{-1}[H]$ is
Borel measurable in $X$ for every open set $H\subseteq Y$.

{\bf(a)} Can there be metric spaces
$X$ and $Y$ and a lower Borel measurable $R\subseteq X\times Y$,
with compact vertical sections, which has no Borel measurable selector?
(\pound8.)   (A negative answer to {\bf CA} would settle this.)

{\bf(b)} Can there be complete metric spaces
$X$ and $Y$ and a lower Borel measurable $R\subseteq X\times Y$,
with closed separable vertical sections, 
which has no Borel measurable selector?
(\pound8.)   (Under certain special axioms, 'no';  see {\smc Fremlin 87},
3N.)

\medskip

{\bf CE$\dagger$.}  Can there be a cardinal $\kappa$ such
that (i) $\Cal P\kappa$ is not countably generated as $\sigma$-algebra
(ii) whenever $J$ is a set and $\Cal A\subseteq\Cal PJ$ has cardinal
$\kappa$, there is a countable $\Cal B\subseteq\Cal PJ$ such that
$\Cal A$ is included in the $\sigma$-algebra of subsets of $J$
generated by $\Cal B$?   (\pound16.)   
What if $\kappa=\omega_1$?  (\pound16.)

\medskip

{\bf CG.} Is it relatively consistent with ZFC that every universally 
measurable set has the Baire 
property?   ({\smc Mauldin Preiss \& Weizs\"acker 83}.)
%p 985, following Cor 5.2
(\pound4.)

\medskip

{\bf CJ.}  Let $A\subseteq\BbbR^n$ be a random set chosen by a
Poisson point process with parameter $\lambda$, so that

$$\text{Pr}(\#(A\cap E)=k)
=\bover{(\lambda\mu E)^k}{k!}e^{-\lambda\mu E}$$

\noindent whenever $k\in\Bbb N$ and
$E\subseteq\BbbR^n$ has Lebesgue measure $\mu E$.
Let $G$ be the random graph obtained by taking $A$ for the
set of vertices and joining all points of $A$ which are within unit
distance of each other.   What is the probability that $G$ has an
infinite component?

If $n=1$, it is zero.   For $n\ge 2$, there is a critical value
$\lambda_n$ such that if $\lambda<\lambda_n$ then $G$ almost
surely has no infinite component, while if $\lambda>\lambda_n$
then $G$ almost surely does have an infinite component.   
For an estimate of $\lambda_2$ see {\smc Mertens \& Moore 12}.

An equivalent problem is the following.   Imagine a random set
$C\subseteq\BbbR^n$ generated by the following process.
Set $C_0=\{ 0\}$.   Given $C_r$, set
$V_r=\{ y:\rho(y,C_r)<1\}\setminus\bigcup_{i<r}
V_i$.   Let $C_{r+1}$ be a random subset of $V_r$ chosen by
a Poisson point process with parameter $\lambda$, and continue.
Set $C=\bigcup_{r\in\Bbb N}C_r$.   Then

\centerline{Pr($C$ is infinite) $ > 0\iff$ Pr($G$ has an
infinite component)=1,}

\noindent and

\centerline{Exp$^{\text{n}}(\#(C)) <\infty\iff\exists
\enskip r\in\Bbb N$, Exp$^{\text{n}}(\#(C_r))<1
\iff\lambda<\lambda_n$}

\noindent ({\smc Meester \& Roy 96}, \S3.5).   But
if $\lambda=\lambda_n$, what is Pr($C$ is infinite)?
(\pound22).

\medskip

{\bf CL.} A topological space $X$ has the property `AF$_{\infty}$'
 (see {\smc Fremlin 87}) if no non-meager open set in $X$ can be
covered by a totally ordered family of meager sets.   Can
$\{0,1\}^{\kappa}$ have this property for every $\kappa$?  (\pound3.)
(If $\frak{m}=\frak{c}$, no.)

Compare {\bf CQ}.

\medskip

{\bf CN.} For each cardinal $\kappa$ give $\BbbR^{\kappa}$ the box
product
topology, and let $S_{\kappa}$ be the set of injective functions in
$\BbbR^{\kappa}$.

{\bf (a)}  Is $S_{\omega_1}$ comeager in $\BbbR^{\omega_1}$?
(\pound8.)

{\bf (b)}  Is $S_{\frakc}$ meager in $\BbbR^{\frakc}$?  (\pound8.)

\noindent (See {\smc Fremlin n83}.   If $\kappa<\frak c$ then
$S_{\kappa}$ is not meager in $\BbbR^{\kappa}$;  see
{\smc Kolman \& Shelah 98}.)

\medskip

{\bf CQ.} A measure space $(X,\Sigma,\mu)$ has the property
`AF$_{\infty}$' (see {\smc Fremlin 87})
if no non-negligible measurable set can be covered by a totally ordered
family of negligible sets.   Can every
Radon measure space have this property?
(\pound21).  (If $\frak{m}=\frak{c}$, no.)

Recall that an infinite cardinal $\kappa$ is a {\bf precaliber} of a
Boolean algebra $\frak{A}$ if for every family
$\langle a_{\xi}\rangle_{\xi<\kappa}$ in $\frak{A}\setminus\{\tbf{0}\}$
there is a set $\Gamma\in[\kappa]^{\kappa}$ such that
$\langle a_{\xi}\rangle_{\xi\in \Gamma}$ is centered, that is,
$\inf_{\xi\in I}a_i\ne\tbf{0}$ for
every non-empty finite set $I\subseteq\Gamma$.
Now an equivalent formulation of this problem is:  can every regular
uncountable cardinal be a precaliber of every
probability algebra?   (See {\smc Fremlin 08}, \S525.)

Compare {\bf CL}, {\bf DW}.

\medskip

{\bf CR.} ({\smc Erd\H{o}s 74}.)   Let $A\subseteq\Bbb R$ be an
infinite set.   Is there necessarily a set $E\subseteq\Bbb R$, with
strictly positive Lebesgue measure, not including any set similar to
$A$? (\pound19.)   What if $A=\{2^{-n}:n\in\Bbb N\}$?
(\pound18.)   (See {\smc Svetic 01} and the references there.)
%{\smc Falconer 84}, {\smc Komj\'ath 83}, {\smc Bourgain 87},
%{\smc Koloutzakis 97}, {\smc Humke \& Laczkovich 98}.)

\medskip

{\bf CT.} A topological space $X$ has the property
`Borel-AF$_{\omega}$' if $\bigcup\Cal E$ is meager whenever
$\Cal E$ is a point-countable family of meager
subsets of $X$ and $\bigcup\Cal E'$ is Borel in $X$
for every $\Cal E'\subseteq\Cal E$.  (See {\smc Fremlin 87}, \S8.)
Can there be a metric space without this property?   (\pound4.)

\medskip

{\bf CV.} If $\kappa$, $\lambda$ and $\theta$ are cardinals, with
$1\le\theta\le\lambda\le\kappa$, and $t\in [0,1]$, say that
$(\kappa,t)\Rightarrow [\lambda]^{\theta}$ if whenever
$\langle E_I\rangle_{I\in [\kappa]^{\theta}}$ is a family of
Lebesgue measurable subsets of $[0,1]$ and the Lebesgue measure of
every $E_I$ is at least $t$, then there is a $K\in [\kappa]^{\lambda}$
such that $\bigcap_{I\in [K]^{\theta}}E_I\ne\emptyset$.   Write
$t_0(\kappa,[\lambda]^{\theta})$ for inf$\{ t:(\kappa,t)
\Rightarrow [\lambda]^{\theta}\}$.   It is known that
$\bover{3+\sqrt{17}}{12}\ge t_0(\omega,[4]^3)\ge\bover{5}{9}$,
$t_0(\omega_1,[4]^3)=\bover12$,  $t_0(\frak{c},[4]^3)\ge\bover14$,
$t_0(\frak{c}^+,[4]^3)=0$;  see {\smc Fremlin n85}, 
{\smc Chung \& Lu 99}.   The calculation of 
$t_0(\omega,[4]^3)=\lim_{n\rightarrow\infty}t_0(n,[4]^3)$ 
is an old problem of P.Tur\'an.

{\bf (a)} Is $t_0(\omega,[4]^3)$ exactly $\bover59$? (\pound20.)

{\bf (b)} Can $t_0(\frak{c},[4]^3)$ be $\bover14$?  (\pound20.)

{\bf (c)} Can $t_0(\omega_2,[4]^3)$ be $\bover12$?  (\pound20.)

\medskip

{\bf DC.} Let $\Cal F$ be an ultrafilter on $\Bbb N$, and $\kappa$
a cardinal.   $\Cal F$
is a {\bf p($\kappa$)-point} ultrafilter if it is non-principal and
whenever $\Cal A\subseteq\Cal F$ and $\#(\Cal A)<\kappa$ then
there is an $I\in\Cal F$ such that $I\setminus A$ is finite for
every $A\in\Cal A$.  $\Cal F$ is {\bf $(\omega,\kappa)$-saturating} if
whenever $\Cal B\subseteq\Cal P(\Bbb N\times\Bbb N)$
is such that $\#(\Cal B) <\kappa$ and the first projection
$\pi_1[\bigcap\Cal B_0]$ belongs to $\Cal F$ for every non-empty finite
$\Cal B_0\subseteq\Cal B$, then there is a function 
$f:\Bbb N\to\Bbb N$ such that
$\{ n:(n,f(n))\in B\}\in\Cal F$ for every $B\in\Cal B$.
(See {\smc Fremlin 84},
A3D, and {\smc Fremlin \& Nyikos 89};  also {\smc Stepr\= ans 92}.)
Can it be that $\frak{p}=\frak{c}$ and at the same time there is a
p($\frak{c}$)-point
ultrafilter which is not $(\omega,\frak{c})$-saturating?
(\pound8.)   (If $\frak m=\frak{c}$, no.)

\medskip

{\bf DM.} A topological space $(X,\frak{T})$ is {\bf copolish}
if there is a Polish topology $\frak{S}$ on $X$, coarser than
$\frak{T}$, such that every point of $X$ has a $\frak{T}$-neighbourhood
base consisting of $\frak{S}$-closed sets.   $(X,\frak{T})$ is
{\bf pseudo-complete} if there is a sequence
$\langle\Cal U_n\rangle_{n\in\Bbb N}$ of $\pi$-bases for $\frak{T}$
such that $\bigcap_{n\in\Bbb N}U_n\ne\emptyset$
whenever $\langle U_n\rangle_{n\in\Bbb N}$ is a sequence such that
$\overline{U}_{n+1}\subseteq U_n\in\Cal U_n$ for every
$n\in\Bbb N$.   (See {\smc Fremlin 84}, \S43.)   Is every copolish space
pseudo-complete? (\pound25.)

\medskip

{\bf DN.} Can it be that for every hereditarily
Lindel\"of compact Hausdorff space $X$ there are a metric space $Z$ and
a continuous
function $f:X\rightarrow Z$ such that $\#(f^{-1}[\{z\}])\le 2$ for every
$z\in Z$?  (\pound26.)   (See {\smc Fremlin 84}, 44Qc,
and {\smc Fremlin n86a}, 9C.)

\medskip

{\bf DO.}  Let $\Cal B$ be the algebra of Borel subsets of $[0,1]$ and
$\mu$ the restriction of Lebesgue linear measure to $\Cal B$.

{\bf (a)} If $\frak{c}\ge\omega_3$ can $\mu$ have a lifting? (\pound28.)
(If we add $\omega_2$ Cohen reals to a model
of GCH, then $\mu$ has a lifting;  see {\smc Fremlin 08}, 554I.)

{\bf (b)} Assume that $\frak{m}=\frak{c}>\omega_1$.   Does
$\mu$ have a lifting?  (\pound28.)

{\bf (c)} (A.H.Stone.) Can there be a
lifting $\theta$ of $\mu$ such that the Borel classes
of $\theta E$, as $E$ runs over $\Cal B$, are bounded in $\omega_1$?
(\pound26.)

{\bf (d)} Can there be a lifting $\theta$ of
$\mu_2$, the restriction of Lebesgue
planar measure to the Borel $\sigma$-algebra of $[0,1]^2$ which
{\bf respects coordinates}, i.e., is
such that if $E$, $F\in\Cal B$ then $\theta(E\times F)$ is of the
form $E'\times F'$?  (\pound22.)   (Compare {\smc Talagrand 82} and
{\smc Fremlin 02}, \S346.)
A negative answer might contribute to a solution of {\bf D} above.)

{\bf (e)} If we add $\omega_2$ random reals to a model of GCH, does
$\mu$ have a lifting?  (\pound12.)

\medskip

{\bf DP.} For any $\epsilon>0$, let $\Bbb P_{\epsilon}$ be the
set of open subsets of $[0,1]$ of Lebesgue measure strictly less
than $\epsilon$, and let $\frak{A}_{\epsilon}$ be the regular open
algebra of $\Bbb P_{\epsilon}$ when $\Bbb P_{\epsilon}$ is given
its up-topology (see {\smc Fremlin 84}, 12B-C).   Let $\Bbb P$
be the set of open subsets of $\Bbb R$ of Lebesgue measure
strictly less than 1, and $\frak{A}$ the regular open algebra of
$\Bbb P$ for its up-topology.   It is known that any of these Boolean
algebras can be embedded into all the others, and that the
$\frak{A}_{\epsilon}$, for $0<\epsilon<1$, are
isomorphic ({\smc Truss 88}, or {\smc Fremlin 08}, \S528).   But are
the $\frak A_{\epsilon}$ actually isomorphic to $\frak{A}$?  (\pound15.)

\medskip

{\bf DS.} Assume that $\frak{m}=\frak{c}=\omega_3$.   Can there be a set
$A\subseteq\Bbb R$ of cardinal $\omega_2$
such that $\Bbb R\setminus A$ is not expressible as the union
of $\omega_1$ Borel sets?   (\pound5.)   (See
{\smc Fremlin \& Jasi\'nski 86}.)

\medskip

{\bf DU$\dagger$.} Suppose that $S\subseteq [\omega_1]^{<\omega}$ is
such that for every finite $I\subseteq\omega_1$ there is a
$J\subseteq I$ with $\#(J)\ge\bover12\#(I)$ and $\Cal P J\subseteq S$.

{\bf (a)} Must there be an infinite $A\subseteq\omega_1$ such that
$[A]^{<\omega}\subseteq S$?
(\pound17.)

{\bf (b)} Assume that $\frak{m} >\omega_1$.   Must there be an
uncountable $A\subseteq\omega_1$ such that
$[A]^{<\omega}\subseteq S$?  (\pound16.)

\medskip

{\bf DW.} Can it be that
whenever $\kappa$ is an infinite cardinal, $(X,\mu)$ is a probability
space, $\langle E_{\xi}\rangle_{\xi<\kappa}$ is a family of measurable
subsets of $X$, and
$0\le\gamma<\text{inf}_{\xi<\kappa}\mu E_{\xi}$,
there is a set $\Gamma\in [\kappa]^{\kappa}$ such that
$\mu(\bigcap_{\xi\in I}E_{\xi})\ge\gamma^{\#(I)}$ for every non-empty
finite $I\subseteq\Gamma$?   (\pound8.)   (If {\bf CQ} has a
negative answer, no.  See {\smc Fremlin 88}.)

\medskip

{\bf DX.} Can there be a
countably-generated $\sigma$-subalgebra $\Sigma$ of $\Cal P\omega_1$
such that (i)
$\Cal I=\Sigma\cap\text{NS}_{\omega_1}$ is cofinal with
NS$_{\omega_1}$, the ideal of non-stationary subsets of $\omega_1$ (ii)
the quotient
algebra $\Sigma/\Cal I$ is ccc?  (\pound7.)   (See {\smc Fremlin n86b}.)

\medskip

{\bf DY.} For partially ordered sets $P$, $Q$ say that $P\prT Q$ if
there is a function $f:P\rightarrow Q$ such that
$f[A]$ is unbounded above in $Q$ whenever $A\subseteq P$ is non-empty
and unbounded above in $P$ ($f$ is a `Tukey function';  see
{\smc Fremlin 93b}).

{\bf (a)} Suppose that $(X,\mu)$ is a Radon probability space
of Maharam type $\kappa$.   Write $\Cal N_{\mu}$ for the ideal
of $\mu$-negligible sets, $\Cal N$ for the ideal of Lebesgue negligible
subsets of $\Bbb R$.   Is it necessarily true that
$\Cal N_{\mu}\prT\Cal N\times[\kappa]^{\le\omega}$?  (\pound11.)   (If
V=L, yes;  see {\smc Fremlin 93b}.)

\medskip

{\bf DZ.(a)} [P.J.Nyikos] Can there be a
Radon compact Hausdorff space (see {\bf W} above) which is not
sequentially compact?   (\pound24.)  (If $2^\frak{t}>\frak{c}$, no.)

{\bf (b)} Is the continuous image of a Radon
compact Hausdorff space always Radon?  (\pound24.)

%maybe I can do this \query, using Wage's construction?

\medskip

{\bf EB.} [V.Bergelson]  Let $(\frak{A},\mu)$ be a probability
algebra and $\pi:\frak{A}\rightarrow\frak{A}$ a measure-preserving
automorphism.   Suppose that $a\in\frak{A}$ and that
$\delta<(\mu a)^2$.   Is there necessarily a set $I\subseteq\Bbb N$,
not of zero asymptotic density, such that
$\mu(\pi^i(a)\cap\pi^j(a))\ge\delta$ for all $i$, $j\in I$? (\pound22.)

\medskip

{\bf EC.} For any topological space $X$, write $\Cal K(X)$
for the family of compact subsets of $X$.
Can there be is an analytic, non-Borel
set $X\subseteq\Bbb R$ such that $\Cal K(X)$
is a PCA (= $\Sigma^1_2$) subset of $\Cal K(\Bbb R)$, where
$\Cal K(\Bbb R)$ is given its usual Polish topology?
(If the axiom of projective
determinacy is true, no;  see {\smc Fremlin 90}.)

\medskip

{\bf ED.}  Can there be $\kappa$, $\Cal I$, $\Cal J$, $B$ such that
(i) $\kappa>\omega$ is a cardinal
(ii) $\Cal I$ is a $\kappa$-additive ideal of $\Cal P\kappa$
(iii) $\Cal J$ is a
$\sigma$-ideal of the algebra $\Cal B$ of Borel subsets of $\Bbb R$
(iv) $B$ is a Borel subset of $\BbbR^2$
(v) $\Cal J$ consists precisely
of those Borel subsets of $\Bbb R$ which are included in vertical
sections of $B$
(vi) $\Cal P\kappa/\Cal I\cong\Cal B/\Cal J$ as Boolean algebras
(vii) $\Cal P\kappa/\Cal I$ is atomless, ccc and not $\{0\}$?
(\pound3.)
(If we omit (v), yes, it seems;  see
{\smc G\l\'owczy\'nski 91}.  $\Cal J$ cannot be the Lebesgue null ideal
nor the ideal of meager subsets of $\Bbb R$;  see
{\smc Gitik \& Shelah 89} and {\smc Fremlin 08}, \S547.)

\medskip

{\bf EG.} Assume that $\kappa$ is an atomlessly-measurable cardinal,
that is, that there is an atomless $\kappa$-additive probability $\nu$
with domain $\Cal P\kappa$.

{\bf (b)} Can we be sure that there is a set $A\subseteq\Bbb R$, of
cardinal $\kappa$, such that no uncountable subset
of $A$ is Lebesgue negligible?  (\pound14.)
(Such a set can be found of any cardinal less than $\kappa$;
see {\smc Fremlin 08}, 544G.)

{\bf (d)} Write $\frak{d}$ for cf$(\Bbb N^{\Bbb N})$.   Can $\kappa$ be
less than $\frak{d}$?  (\pound24.) (It is not
possible to have $\kappa=\text{cf}(\frak{d})$;  see 
{\smc Fremlin 08}, 544Nc.)

{\bf (e)} Can there be an atomless
$\kappa$-additive probability $\nu$ with domain $\Cal P\kappa$ such
that the measure algebra $\Cal P\kappa/\Cal N_{\nu}$ is not
homogeneous? (\pound19.)   What if $\kappa=\frak c$?  (\pound8.)

{\bf (f)} Can it be that $\kappa<\frak{c}$
and that $\kappa$ is not weakly $\Pi^1_1$-{\hskip0.001em}indescribable?
(\pound3.) (See {\smc Fremlin 93a}, 4Lb.)

{\bf (g)} Can it be that $\kappa$ is weakly
$\Pi^1_1$-indescribable, but that there is some $n\in\Bbb N$ such that
$\kappa$ is not weakly $\Pi^1_n$-indescribable?   (See {\smc Fremlin
93a}, 4R.) (\pound1.)

{\bf (h)} Suppose that $\nu$ is {\bf normal}, that is, that whenever
$\langle E_{\xi}\rangle_{\xi<\kappa}$ is a family of subsets of $\kappa$
such that $\nu E_{\xi}=1$ for every $\xi$, then
$\nu\{\xi:\xi<\kappa,\,\xi\in E_{\eta}\enskip\forall\enskip\eta<\xi\}=1$.
Let $\langle A_{\zeta}\rangle_{\zeta<\omega_1}$ be a family of subsets of
$\kappa$ such that $\nu A_{\zeta}>0$ for every $\zeta<\omega_1$.   Must
there be a countable set $N\subseteq\kappa$ such that
$N\cap A_{\zeta}\ne\emptyset$ for every $\zeta<\omega_1$?   
(See {\smc Fremlin 08}, 555F.)(\pound15.)

{\bf (i)} Can it be that every infinite compact
Hausdorff space of weight less than $\kappa$ has a non-trivial
convergent sequence?  (\pound6.  See {\smc Dow \& Fremlin 07}.)

\medskip

{\bf EI.} (T.Bartoszy\'nski.) For Boolean algebras $\frak{A}$,
$\frak{B}$ write $\frak{A}\otimes\frak{B}$ for their free product
(see {\smc Fremlin 02}, \S315), and $\frak{A}\hat\otimes\frak{B}$
for the Dedekind completion of $\frak{A}\otimes\frak{B}$.   Now take
$\frak{A}$ to be the Lebesgue measure algebra.   Is
$\frak{A}\hat\otimes\frak{A}$ isomorphic to
$\frak{A}\hat\otimes\frak{A}\hat\otimes\frak{A}$?  (\pound11.)
Is there an order-continuous Boolean homomorphism from
$\frak{A}\hat\otimes\frak{A}\hat\otimes\frak{A}$ to
$\frak{A}\hat\otimes\frak{A}$?  (\pound11.)

\medskip

{\bf EJ.} A {\bf measurable space with negligibles} is a triple
$(X,\Sigma,\Cal I)$ where $X$ is a set, $\Sigma$ is a $\sigma$-algebra
of subsets of $X$, and $\Cal I$ is a $\sigma$-ideal of $\Cal P X$
generated by $\Sigma\cap\Cal I$.   (See {\smc Fremlin 87}.)
In this case, $(X,\Sigma,\Cal I)$ is {\bf $\omega_1$-saturated}
if $\Sigma/\Sigma\cap\Cal I$ is ccc, and {\bf complete} if
$\Cal I\subseteq\Sigma$;  a {\bf lifting} of
$(X,\Sigma,\Cal I)$ is a Boolean homomorphism
$\theta:\Sigma\to\Sigma$ such that $\theta E=\emptyset$ whenever
$E\in\Sigma\cap\Cal I$
and $E\triangle\theta E\in\Cal I$ for every $E\in\Sigma$.   (See {\bf D}
above.)

{\bf (b)} (A.~H.~Stone) Taking $X=\Bbb R$, $\Cal B$ the algebra of
Borel subsets of $\Bbb R$ and $\Cal M$ the ideal of meager subsets of
$\Bbb R$,  can there be a
lifting $\theta$ of $(\Bbb R,\Cal B,\Cal M)$ such that the Baire classes
of $\theta E$, as $E$ runs over $\Cal B$, are bounded in $\omega_1$?
(\pound7.) (See {\bf DO(c)}.)

\medskip

{\bf EK.} The following questions seem to be left over from {\smc
Talagrand 84}
and {\smc Shelah \& Fremlin 93}, in addition to {\bf R} and {\bf AD}
above.

{\bf (a)} Write $\eusm L^0(\Sigma)$ for the space of real-valued
Lebesgue measurable
functions on $[0,1]$ and $\frak T_p$ for the pointwise topology of
$\BbbR^{[0,1]}$.   Can there be a $\frak T_p$-separable
$\frak T_p$-compact set $K\subseteq\eusm L^0(\Sigma)$ such that the
closed convex
hull of $K$ in $\BbbR^{[0,1]}$ does not lie within $\eusm L^0(\Sigma)$?
(\pound12.)

{\bf (b)} Can there be a separable compact
Radon measure space $(X,\frak T,\Sigma,\mu)$ and a function
$f:[0,1]\times X\to\Bbb R$ which is measurable in the first variable,
continuous in the
second, but not jointly measurable with respect to the completed product
measure on $[0,1]\times X$?   (\pound6.)   (A positive answer to
(a) would imply the same answer to (b).)

{\bf (c)} Can there be a set
$A\subseteq[0,1]$, of outer Lebesgue measure 1, and a
$\frak T_p$-compact subset $K\subseteq\BbbR^A$ such that every function
in $K$ is measurable for the subspace measure on $A$, but not every
sequence in $K$ has a subsequence convergent almost everywhere?
(\pound8.)
(If $\frak m=\frak c$, no;  see {\smc Talagrand 84}, 9-3-3.)

\medskip

{\bf EL. (b)} For which cardinals $\kappa$ is it always true that whenever
$\Cal A$ is a family of subsets of $[0,1]$ and $\#(\Cal A)\le\kappa$
then there is a set $B\subseteq[0,1]$ such that
$\mu_L^*(A\cap B)=\mu_L^*(A\setminus B)=\mu_L^*A$ for every
$A\in\Cal A$, writing $\mu_L$
for Lebesgue measure?   This is true for $\kappa\le\omega$ 
and possibly false for
$\kappa=\frak c$ (see {\smc Fremlin 08}, \S547).
What about $\kappa=\omega_1$?  (\pound22.)

{\bf (c)} Repeat (b) for category in the place of measure.

\medskip

{\bf EN.} Let $(X,\mu)$ be a probability space and $E_0,\ldots,E_n$
stochastically independent
measurable subsets of $X$ with $\sup_{i\le n}\mu E_i=\epsilon$.
Suppose that $E\subseteq X$ is a measurable set such that 
$E\times E\subseteq\bigcup_{i\le n}E_i\times E_i$, and that
$\epsilon\le{1\over 2}$.
Then $\mu E\le\epsilon$ ({\smc Fremlin 91}).   But suppose that
$E\times E\times E\subseteq\bigcup_{i\le n}E_i\times E_i\times E_i$
and that $\epsilon\le{2\over 3}$.   Does it now follow that
$\mu E\le\epsilon$? (\pound16.)

\medskip

{\bf ER.} Let $\nu$ be {\bf Talagrand's measure} on $\Cal P\Bbb N$, that
is, for $E\subseteq\Cal P\Bbb N$ and $\alpha\in[0,1]$, $\nu E$ is
defined and equal to $\alpha$ iff there are $H\subseteq\Cal P\Bbb N$,
$\langle\Cal F_n\rangle_{n\in\Bbb N}$ such that (i) $\mu H$ exists and
is equal to $\alpha$,
where $\mu$ is the usual Radon measure on $\Cal P\Bbb N$ (ii) each 
$\Cal F_n$ is a filter on $\Bbb N$ with $\mu^*\Cal F_n=1$ (iii)
$E\cap\bigcap_{n\in\Bbb N}\Cal F_n=H\cap\bigcap_{n\in\Bbb N}\Cal F_n$.
Then the map
$a\mapsto\chi a:\Cal P\Bbb N\to\ell^{\infty}$ is scalarly measurable,
that is, $a\mapsto h(\chi a)$ is $\dom\nu$-measurable for every
$h\in(\ell^{\infty})^*$.
(See {\smc Fremlin \& Talagrand 79}, \S1, {\smc Talagrand 84}, 13-3-2 or
{\smc Fremlin 03}, \S464.)   But is it measurable for
the weak topology $\frak T_s(\ell^{\infty},(\ell^{\infty})^*)$, in the
sense that for any weakly open set $G\subseteq\ell^{\infty}$ the set
$\{a:\chi a\in G\}$ is measured by $\nu$? (\pound19.)

\medskip

{\bf ET$\dagger$.} Let $\langle G_x\rangle_{x\in[0,1]}$ be a family of
open subsets of $\Bbb R$ such that
$\mu(A\setminus\bigcup_{x\in A}G_x)=0$ for
every $A\subseteq[0,1]$, where $\mu$ is Lebesgue measure on $\Bbb R$.
Let $\epsilon>0$.   Is there necessarily
a family $\langle H_x\rangle_{x\in[0,1]}$ of open sets such that
$x\in H_x$ for every $x\in [0,1]$ and 
$\mu(\bigcup_{x\in A}H_x\setminus\bigcup_{x\in A}G_x)\le\epsilon$ 
for every $A\subseteq[0,1]$?
(\pound17.)  (This is relevant to Problem 4Ga in {\smc Fremlin 95}.)

\medskip

{\bf EU.} Let $\sequencen{(X_n,\mu_n)}$ be a sequence of probability
spaces, with
(completed) product $(X,\Sigma,\mu)$.   Is there necessarily a lifting
$\theta$ for $\mu$ which respects coordinates in the sense that
$\theta E$ is determined by coordinates in $J$ whenever $E\in\Sigma$ is
determined by coordinates in $J\subseteq\Bbb N$?
(\pound10.)   (If every $\mu_n$ is Maharam-type-homogeneous,
yes;  see {\smc Fremlin 02}, \S346.)
What if $X_n=\{0,1\}$, $\mu_n\{1\}=\bover23$ for every $n$?  (\pound9.)

\medskip

{\bf EV.} If $(X,\Sigma,\mu)$ is a measure space, a {\bf lower density}
for $\mu$ is a function $\phi:\Sigma\to\Sigma$ such that
(i) $\phi\emptyset= \emptyset$
(ii) $\phi(E\cap F)=\phi E\cap\phi F$ for all $E$, $F\in\Sigma$
(iii) $\phi E=\phi F$ whenever $\mu(E\symmdiff F)=0$
(iv) $\mu(E\symmdiff\phi E)=0$ for every $E\in\Sigma$.
Let $\mu$ be the usual measure on
$X=\{0,1\}^{\Bbb N}$.   Is there a lower density for $\mu$ which is
invariant under permutations of coordinates?   (\pound16.)

\medskip

{\bf EY.} Take $p\in\ooint{1,\infty}$ and $f$, $g:[0,1]\to\Bbb R$
measurable functions such that $\alpha=(\int|f|^p)^{1/p}$,
$\beta=(\int|g|^p)^{1/p}$ are finite and the conditional expectation of
$g$ on $f$ is zero.   For given $\alpha$, $\beta$ what is the greatest
possible value of $(\int|f+g|^p)^{1/p}$?   (\pound1.)
The case of principal interest is $\beta\le\alpha$.

\medskip

{\bf FA.} Can it be that every
totally finite completion regular Borel measure on a completely regular
Hausdorff space is $\tau$-additive?   (\pound14.)
(If {\it either} there is a real-valued-measurable cardinal {\it or}
$\frak c=\omega_1$, no;  see {\smc Fremlin 03}, 439I, and
{\smc Juh\'asz Kunen \& Rudin 76}.)

\medskip

{\bf FB.} ({\smc Wajch 91}) Let $\frak A$ be a Boolean algebra.
A {\bf Maharam submeasure} on $\frak A$ is a functional 
$\nu:\frak A\to\coint{0,\infty}$ such that $\nu 0=0$,
$\nu a\le\nu(a\Bcup b)\le\nu a+\nu b$ for all $a$, $b\in\frak A$, and
$\lim_{n\to\infty}\nu a_n=0$ whenever $\sequencen{a_n}$ is a non-increasing
sequence in $\frak A$ with infimum $0$.
If $\nu$ is a Maharam submeasure with domain
$\Cal PX$, must there be a measure $\mu$, with domain $\Cal PX$, such
that $\nu E=0$ whenever $\mu E=0$?  (\pound13.)
(If there is no `quasi-measurable cardinal' ({\smc Fremlin 08}, \S542), 
yes.)

\medskip

{\bf FC.} Let $\mu$ be Lebesgue measure on $\Bbb R$, and $\nu$ `Cantor
measure', that is, the image of the usual measure on $\{0,1\}^{\Bbb N}$
under the map $x\mapsto 2\sum_{i=0}^n3^{-i-1}x(i)$.   Must there be a
set $A\subseteq\Bbb R$ such that $\mu^*A>0$ but $\nu(x+A)=0$ for every
$x\in\Bbb R$?   (\pound11.)
(If the uniformity of Lebesgue measure is $\frak c$, 
or the covering number of Lebesgue measure is equal to the cofinality of 
Lebesgue measure, yes;  and if we replace 
ordinary Cantor measure by `middle three-fifths' Cantor measure, yes; 
see {\smc
Fremlin n09}.   There is no such Borel set;  see {\smc Fremlin 03},
444L.)  

\medskip

{\bf FD.} Let $(X,\frak T,\Sigma,\mu)$ be a Radon probability space, of
countable Maharam type, such that $\mu G>0$ for every non-empty open set
$G\subseteq X$.   Does it necessarily have a lifting $\phi$ which is
{\bf strong}, in the sense that $\phi G\supseteq G$ for every open set
$G$?   (\pound11.)
(If $\frak c=\omega_1$, yes;  see {\smc Mokobodzki 75} or
{\smc Fremlin 08}, \S535.   For a Radon measure with no
strong lifting, see {\smc Losert 79} or {\smc Fremlin 03}, 453N.)

\medskip

{\bf FF. (a)} Let $X$ and $Y$ be topological spaces with Borel
probability measures $\mu$ and $\nu$.   Is there necessarily a Borel
measure $\lambda$ on $X\times Y$ such that
$\lambda(E\times F)=\mu E\cdot\nu F$ for all Borel sets $E\subseteq X$,
$F\subseteq Y$?   (\pound12.)
(See {\smc Fremlin 03}, 435O.)

{\bf (b)} A {\bf Baire measure} on a topological space is a measure with
domain the Baire $\sigma$-algebra (see {\bf D} above).   Let $X$ and $Y$
be completely regular Hausdorff spaces with
Baire probability measures $\mu$ and $\nu$.   Is there necessarily a
Baire measure $\lambda$ on $X\times Y$ such that
$\lambda(E\times F)=\mu E\cdot\nu F$ for all Baire sets $E\subseteq X$,
$F\subseteq Y$?  (\pound11.)

\medskip

{\bf FG.} Can it be that whenever $\mu$ is a Baire
probability measure on a normal Hausdorff space $X$, then $\mu$ has an
extension to a Borel measure?   (\pound13.)
(If $\clubsuit$, no.  Only Dowker spaces $X$ need be considered.  See
{\smc Fremlin 03}, 435C and 439O, and {\smc Kojman \& Michalewski p06}.)

\medskip

{\bf FJ.} (M.Foreman) {\bf (a)} Let us say that a filter $\Cal F$ on
$\Bbb N$ is
{\bf measure-converging} if whenever $\sequencen{f_n}$ is a sequence of
measurable functions on a probability space (equivalently, on $[0,1]$)
converging to $0$ in measure, then $\lim_{n\to\Cal F}f_n(x)=0$ for
almost every $x$.   (See {\smc Fremlin 08}, \S538.)   Must there be a
measure-converging filter on $\Bbb N$?   (\pound14.)
(If $\frak p=\frak c$, yes.)
%Then every rapid filter is measure-converging (because there is a set
%$F\in\Cal F$ such that
%$\sum_{n\in F}\int\min(1,|f_n|)<\infty$).
%Talagrand has shown that if there is a rapid filter, there is a
%measure-converging filter which is not rapid.
%(If $\Cal F$ is rapid, let $\Cal G$ be the filter
%$\{a:\lim_{n\to\Cal F}2^{-n}\#(a\cap 2^{n+1}\setminus 2^n)=1$.)

{\bf (b)} Every rapid filter is measure-converging.
(A filter $\Cal F$ is {\bf rapid} if for every sequence
$\sequencen{x_n}$ of positive numbers converging to $0$ there is an
$F\in\Cal F$ such that $\sum_{n\in F}x_n$ is finite.)
If there is a measure-converging filter, must there be a rapid
filter?   (\pound6.)

\medskip

{\bf FK.} Are there a Hausdorff space $X$ and a transitive group $G$
of autohomeomorphisms of $X$ and two non-zero $G$-invariant Radon
measures on $X$ which are not scalar multiples of each other?   (If $X$
is locally compact and metrizable, `no';  see {\smc Fremlin n00}.)
(\pound9.)
What if $X$ is compact? (\pound9.)
%if  X =\BbbR^2 with separately-continuous topology, there is an example
%with \mu inner regular wrt compacta, but not loc finite

\medskip

{\bf FL.} Let $\frak A$ be the Lebesgue measure algebra, $\Aut(\frak A)$
the group of all Boolean algebra automorphisms of $\frak A$,
$\Aut_{\mu}(\frak A)$ the group of measure-preserving automorphisms of
$\frak A$.   Can either $\Aut(\frak A)$ or $\Aut_{\mu}(\frak A)$ be
generated (as a group) by finitely many abelian subgroups?   (\pound6.)
%Abert & Keleti:  yes if $\frak A=\Cal P\kappa$

\medskip

{\bf FM.} Assume that $\frak m>\omega_1$.   Let $X$ be a Banach space of
weight $\omega_1$.   Must $X$ be a Radon space when given its weak
topology?   (\pound10.)

\medskip

{\bf FN$\dagger$.} A family $\Cal K$ of sets is countably compact if any
sequence in $\Cal K$ with the finite intersection property
has non-empty intersection.    A measure space $(X,\Sigma,\mu)$ is
countably compact if $\mu$ is inner regular with respect to some
countably compact family of sets.   Suppose now that $\mu$ is a
probability measure defined on a $\sigma$-subalgebra of the Borel
algebra of $[0,1]$.   Must $\mu$ be countably compact?   (\pound10.)
(If $\frak c=\omega_1$, yes.)

\medskip

{\bf FO.} For a family $\Cal K$ of sets, write $\Gamma(\Cal K)$ for the
infinite game in which players choose alternately elements of $\Cal K$,
each a subset of the preceding one, and the second player wins if the
intersection of the sequence is non-empty.   $\Cal K$ is {\bf weakly
$\alpha$-favourable} if the second player has a winning strategy in
$\Gamma(\Cal K)$, and {\bf $\alpha$-favourable} if he has a winning
tactic, that is, a winning strategy depending only on the first player's
previous move.   Is there a probability space $(X,\Sigma,\mu)$ such that
$\Sigma\setminus\Cal N_{\mu}$ is weakly $\alpha$-favourable but not
$\alpha$-favourable, where $\Cal N_{\mu}$ is the ideal of
$\mu$-negligible sets?   (\pound11.)  (See {\smc Fremlin 00}.)

\medskip

{\bf FP.} Let $P$ be a partially ordered set.   A {\bf Freese-Nation
function} on $P$ is a function $f:P\to\Cal PP$ such that
$f(p)\cap f(q)\cap[p,q]\ne\emptyset$ whenever $p\le q$ in $P$.   The
{\bf Freese-Nation number} $\FN(P)$ of $P$ is the smallest cardinal
$\kappa$ such that there is a Freese-Nation function from $P$ to
$[P]^{<\kappa}$.

(a) For finite $n$, how large is the smallest $P$ such that $\FN(P)=n$?
(\pound3.)(For $n\ge 4$, it has between $2n-4$ and $4n-11$ elements.)

(b) Set $f(n)=\FN(\Cal Pn)-1$.   Then $f(3)=4$ and
$\lim_{n\to\infty}f(n)^{1/n}=\inf_{n\ge 1}f(n)^{1/n}$ lies between
$2/\sqrt{3}$ and $\root 3 \of 4$ ({\smc Fremlin \& Penman p01}).   What
is its exact value?
(\pound2.)

\medskip

{\bf FR$\dagger$.} Can it be that whenever $X$ is a compact Hausdorff
space and $\familyiI{\mu_i}$ is a family of mutually singular Radon
probability measures on $X$, then $\#(I)\le\#(X)$?   (\pound3.)

\medskip

{\bf FU.} Let $G$ and $H$ be Polish groups (that is, topological groups
for which the topologies are Polish) and $f:G\to H$ a homomorphism which
is universally measurable
(that is, the inverse image of any open set in $H$ is a universally
measurable subset of $G$).   Does it follow that $f$ is continuous?
(If either group is abelian, yes;  see {\smc Christensen 74}, chap.\ 7.)
(\pound9.)

\medskip

{\bf FV.} (See {\smc Fremlin 08}, \S534.) (a) Let $\CalSmz$ be the
family of subsets of
$\Bbb R$ with strong measure zero.   Can we prove in ZFC that
$\cf\CalSmz\ge\omega_2$?   (\pound4.)

(b) Two metric spaces $(X,\rho)$ and $(Y,\sigma)$ are 
{\bf $\CalSmz$-equivalent} if there is a bijection $f:X\to Y$ such that 
a set
$A\subseteq X$ has strong measure zero for $\rho$ iff $f[A]$ has strong
measure zero for $\sigma$.    How many types of Polish spaces under
$\CalSmz$-equivalence can there be?   If we give $\BbbZ^{\Bbb N}$ and
$\BbbR^{\Bbb N}$ right-translation-invariant metrics inducing their
topologies, are they (in ZFC) $\CalSmz$-equivalent?   (\pound7.)

(c) Suppose that there is a separable metric space
$(X,\rho)$ of cardinal $\frak c$ with strong measure zero.   Is there
necessarily a subset of $\Bbb R$, of cardinal $\frak c$, with strong
measure zero?   (\pound5.)

\medskip

{\bf FX.} A filter $\Cal F$ on $\Bbb N$ is {\bf measurable} if it is
measured by the usual measure on $\Cal P\Bbb N$ (in which case it must
be negligible, by the zero-one law).   Must there be a measurable filter
which is the intersection of $\omega_1$ ultrafilters on $\Bbb N$?
(\pound5).   (If $\frak c=\omega_1$, yes.   See {\smc Fremlin 03},
\S464.)

\medskip

{\bf FY.} Let $X$ be the set of all partial orders on $\Bbb N$, regarded
as a closed subspace of the compact Hausdorff space
$\Cal P(\Bbb N\times\Bbb N)$.   Let $G$ be the group of all permutations
of $\Bbb N$, and for $a\in G$, $x\in X$ write
$a\action x=\{(i,j):(a^{-1}(i),a^{-1}(j))\in x\}$, so that $\action$ is
a continuous action of $G$ on $X$ if $G$ is given its topology of
pointwise convergence.   Let $Q$ be the set of $G$-invariant Radon
probability measures on $X$.   What are the extreme points of $Q$?
(\pound8.)

\medskip

{\bf FZ.} Can $\FN(\Cal P\Bbb N)$ (definition:  {\bf FP} above) have
countable cofinality?  (\pound7.)

\medskip

{\bf GA.} Can there be a hereditarily Lindel\"of compact Hausdorff space
$X$ with a Radon probability measure of Maharam type $\omega_2$?  
(\pound7.)(If
{\it either} $\frak c=\omega_1$ {\it or} $\Bbb R$ is not covered by
$\omega_1$ Lebesgue negligible sets, no.   See {\smc Fremlin 08}, \S531.)

\medskip

{\bf GB.} If $\Cal I$ is a proper ideal of subsets of a set $X$, its
{\bf shrinking number} $\shr\Cal I$ is the least cardinal $\kappa$ such
that for every $A\in\Cal PX\setminus\Cal I$ there is a
$B\in[A]^{\le\kappa}\setminus\Cal I$.   Now suppose that $\mu$ is
Lebesgue measure on $[0,1]$ and that $(Z,\nu)$ is the Stone space of the
measure algebra of $([0,1],\mu)$ ({\smc Fremlin 02}, 321K).   Let
$\Cal N(\mu)$, $\Cal N(\nu)$ be the respective null ideals.   Can
$\shr\Cal N(\mu)$ and $\shr\Cal N(\nu)$ be different?
(\pound7.)

\medskip

{\bf GC.} (S.Zeberski) Let $\Cal A$ be a partition of $[0,1]$ into
Lebesgue negligible sets.   Must there be a $\Cal C\subseteq\Cal A$ such
that $\bigcup\Cal C$ has inner measure 0 and outer measure 1?
(See {\smc Fremlin \& Todor\v{c}evi\'c n04}.)   (\pound7.)
If every member of
$\Cal A$ is countable, yes ({\smc Kumar \& Shelah p15}).


\medskip

{\bf GD.} If $\Cal I$ is a proper ideal of subsets of a set $X$ with
$\bigcup\Cal I=X$, then $\add\Cal I$ is the least cardinal of any subset
of $\Cal I$ with union not in $I$, and $\cov\Cal I$ is the least
cardinal of any subset of $\Cal I$ with union $X$.   For a cardinal
$\kappa$, write $\Cal N_{\kappa}$ for the null ideal of the usual
measure on $\{0,1\}^{\kappa}$.   (See {\smc Fremlin 08}, \S521.)
Can $\add\Cal N_{\omega}$ be greater than $\cov\Cal N_{\omega_1}$?
(\pound6.)

\medskip

{\bf GE$\dagger$.} A Boolean algebra $\frak A$ is {\bf weakly 
$\sigma$-distributive} or {\bf $\omega^{\omega}$-bounding} 
if for every sequence $\sequencen{A_n}$ of countable partitions of unity 
in $\frak A$ there is a
partition $C$ of unity such that $\{a:a\in A_n$, $a\Bcap c\ne 0\}$ 
is finite for every $n\in\Bbb N$, $c\in\frak C$.    
Let $\Cal B$ be the Borel $\sigma$-algebra of $[0,1]$, and $\Cal I$ a 
$\sigma$-ideal of $\Cal B$ such that for every 
$E\in\Cal B\setminus\Cal I$ there is a compact set $K\subseteq E$ such that 
$K\notin\Cal I$.   Is the quotient algebra $\frak A=\Cal B/\Cal I$ 
necessarily weakly $\sigma$-distributive?   (\pound4.)
What if $\frak A$ is ccc?   (\pound4.)

\medskip

{\bf GF.} A Boolean algebra $\frak A$ is {\bf\sfcc} if 
$\frak A\setminus\{0\}$ is expressible as $\bigcup_{n\in\Bbb N}A_n$ where
no $A_n$ includes any infinite disjoint family.   
$\frak A$ is {\bf\sbcc} if 
$\frak A\setminus\{0\}$ is expressible as $\bigcup_{n\in\Bbb N}A_n$ where
no $A_n$ includes any disjoint family of size greater than $n$ members.
Is there a \sfcc\ Boolean algebra which is not \sbcc? (\pound5.)

\medskip
   
{\bf GG.} A {\bf Maharam algebra} is a Dedekind complete Boolean algebra
$\frak A$ such that there is a Maharam submeasure 
$\nu:\frak A\to\coint{0,\infty}$ (definition:  {\bf FB} above) 
which is {\bf strictly positive}, that is, $\nu a>0$ whenever $a\ne 0$.   
A {\bf measurable algebra} is a Dedekind complete Boolean algebra $\frak A$
such that there is a strictly positive countably additive functional
$\mu:\frak A\to\coint{0,\infty}$ (see {\smc Fremlin 02}, chap.\ 39).
Now that we know that we have non-trivial Maharam algebras
({\smc Talagrand 06}, or {\smc Fremlin 02}, \S394), 
we can ask the following questions.

{\bf (a)} If $\frak A$ is a
non-zero atomless Maharam algebra, must it have an atomless order-closed
subalgebra $\frak C$ which is a measurable algebra? (\pound6.)

{\bf (b)} Is every Maharam algebra \sbcc?  (definition:  {\bf GF} above).
(See {\smc Fremlin n06}, \S4.)(\pound6.)

{\bf (c)} Let $\frak A$ be a Maharam algebra and $\nu$ a strictly positive
Maharam submeasure on $\frak A$.   Then for every $\epsilon>0$ we have a
rank function $r_{\epsilon}:\frak A\to\On$ defined by saying that

\Centerline{$r_{\epsilon}(a)
=\sup\{r_{\epsilon}(b):b\subseteq a$, $\nu(a\Bsetminus b)\ge\epsilon\}$}

\noindent for every $a\in\frak A$ ({\smc Fremlin n06}, \S6).   
Is it possible to have $r_{\epsilon}(1)\ge\omega_1$?  (\pound6.)
that $\frak A$ is measurable iff $r_{\epsilon}(1)\le\omega$ for every
$\epsilon>0$;  see {\smc Fremlin n06}, \S7.
%It is known that for every $\xi<\omega_1$ there is a Maharam algebra 
%$\frak A$ with a strictly positive Maharam submeasure $\nu$ for which
%$r_1(1)\ge\xi$;  see {\smc Perovi\'c \& Veli\v{c}kovi\'c p17}.

{\bf (d)} A Boolean algebra $\frak A$ is {\bf$\sigma$-linked} if 
$\frak A\setminus\{0\}$ is expressible as $\bigcup_{n\in\Bbb N}A_n$ where
no $A_n$ contains any pair of disjoint elements.
A measurable algebra is $\sigma$-linked iff it has cardinal at most 
$\frak c$ ({\smc Fremlin 08},  524Mf).   But does a Maharam algebra of
cardinal $\frak c$ have to be $\sigma$-linked?  (\pound5.)

{\bf (e)} Let $\frak A$ be a Boolean algebra.   Its {\bf$\pi$-weight}
$\pi(\frak A)$ is the least cardinal of any coinitial set in 
$\frak A\setminus\{0\}$;  its {\bf centering number} $d(\frak A)$ is the
least cardinal of any cover of $\frak A\setminus\{0\}$ by
downwards-directed sets not containing $0$.
Let $\frak A$ be an atomless Maharam algebra of countable
Maharam type, not $\{0\}$.   
Must we have $\pi(\frak A)=\pi(\frak B_{\omega})$, 
where $\frak B_{\omega}$ is the
measure algebra of Lebesgue measure?  (\pound5.)   Must we have
$d(\frak A)=d(\frak B_{\omega})$?  (\pound5.)    
(See {\smc Fremlin 08}, chap.\ 52, and {\smc Fremlin n06}, \S9.)

{\bf (f)} Is there a rigid Maharam algebra other than $\{0\}$ and
$\{0,1\}$?   (\pound3.)

\medskip

{\bf GH.} A measure space $(X,\Sigma,\mu)$ has the 
{\bf measurable envelope property} 
if every subset $A$ of $X$ has a measurable envelope, that is, a
set $E\in\Sigma$ such that $A\subseteq E$ and any measurable subset of
$E\setminus A$ is negligible ({\smc Fremlin 01}, 213Xl).   Is it possible
that Hausdorff $1$-dimensional measure on $\BbbR^2$ has the
measurable envelope property?   (\pound6.)  (If $\frak m=\frak c$, no;
see {\smc Fremlin n05}.)

\medskip

{\bf GI$\dagger$.} 
A Boolean algebra $\frak A$ is {\bf$\sigma$-$n$-linked} if 
$\frak A\setminus\{0\}$ is expressible as $\bigcup_{k\in\Bbb N}A_k$ where
$\inf_{i<n}a_i\ne 0$ for any family $\family{i}{n}{a_i}$ in any $A_k$.
Is it possible that every Boolean algebra which is
$\sigma$-$n$-linked for every $n$ has centering number less than $\frak c$?
(\pound5.)

\medskip

\leaveitout{{\bf GJ.} (M.Di Nasso) A {\bf Hausdorff} or {\bf Property C} ultrafilter 
on $\Bbb N$ ({\smc Daguenet 79}, or {\smc Fremlin 08}, \S538) is a
non-principal ultrafilter $\Cal F$ on $\Bbb N$ such that whenever $f$,
$g\in\NN$ and $f[[\Cal F]]=g[[\Cal F]]$ then $\{n:f(n)=g(n)\}$ belongs to
$\Cal F$.   Show that it is possible that there are no
Hausdorff ultrafilters.   (\pound6.)

\medskip
}

{\bf GK.} Let $X$ be a Hausdorff space.   A {\bf Radon submeasure} on $X$
is a Maharam submeasure $\nu$
with domain $\Sigma\subseteq\Cal PX$ such that
(i) every open set belongs to $\Sigma$ (ii) 
$\nu E=\sup\{\nu K:K\subseteq E$ is compact$\}$ for every $E\in\Sigma$
(iii) $E\in\Sigma$ whenever $E\subseteq F\in\Sigma$ and $\nu F=0$.
A {\bf lifting} for a submeasure $\nu$ on a Boolean algebra $\frak A$
is a Boolean homomorphism
$\phi:\dom\nu\to\dom\nu$ such that (i) $\phi a=0$ whenever $\nu a=0$ (ii)
$\nu(a\Bsymmdiff\phi a)=0$ for every $a\in\dom\nu$.   Which Radon
submeasures have liftings?   (\pound2.) (See {\smc Fremlin n06}.)

\medskip

{\bf GL.} (R.W.Kaye) Let $\Cal N$ be the Lebesgue null ideal.   Is there a
non-trivial automorphism $\pi$ of the quotient Boolean algebra
$\Cal P\Bbb R/\Cal N$ such that $\pi E^{\ssbullet}=E^{\ssbullet}$ for every
Lebesgue measurable set $E$?  (\pound7.)

\medskip

{\bf GM.} ({\smc Bogachev \& Lukintsova p07}) If $X$ is a topological space
and $\mu$ is a probability measure on $X$, I say that a sequence
$\sequencen{x_n}$ in $X$ is {\bf equidistributed} if 
$\mu F\ge\limsup_{n\to\infty}\Bover1n\#(\{i:i<n$, $x_i\in F\}$ for every
measurable closed set $F\subseteq X$ ({\smc Fremlin 03}, \S491).
Suppose that $X$ and $Y$ are compact Hausdorff spaces such that every Radon
probability measure on either $X$ or $Y$ has an equidistributed sequence.  
Does it follow that every Radon probability measure on $X\times Y$ has an
equidistributed sequence?  (\pound6.)

\medskip

{\bf GO$\dagger$.} (J.Mycielski) 
Let $\mu$ be a Radon probability measure on
$\BbbR^r$, where $r\ge 1$, and $F\subseteq\BbbR^r$ a closed set.   For
$\pmb{x}\in(\BbbR^r)^{\Bbb N}$, $n\in\Bbb N$ and $y\in\BbbR^r$, set
$\delta_{\pmb{x}yn}=\min_{i\le n}\|\pmb{x}(i)-y\|$,
$k_{\pmb{x}yn}=\min\{i:i\le n$, 
$\|\pmb{x}(i)-y\|=\delta_{\pmb{x}yn}\}$;
set $F_{\pmb{x}n}=\{y:\pmb{x}(k_{\pmb{x}yn})\in F\}$.   Is it always the
case that $\lim_{n\to\infty}\mu(F\symmdiff F_{\pmb{x}n})=0$ for
$\mu^{\Bbb N}$-almost every $\pmb{x}$?  (\pound4.)  (If $r=1$, yes;  if
$\mu$ is the uniform measure on $[0,1]^r$, yes.)

\medskip

{\bf GQ.} For $k\in\Bbb N$, say that a topological
group $G$ is {\bf $k$-Steinhaus} if whenever $V\subseteq G$ is a symmetric
set, containing the identity, such that countably many left translates of 
$V$ cover $G$, then $V^k$
is a neighbourhood of the identity.   (See {\smc Rosendal \& Solecki 07}.)
Let $G$ be the group of measure-preserving automorphisms of the measure
algebra $(\frak A,\bar\mu)$ of Lebesgue measure on $[0,1]$;  give it the 
topology generated by the metric $\rho$, where 
$\rho(\pi,\phi)=\sup_{a\in\frak A}\bar\mu(\pi a\Bsymmdiff\phi a)$.   
Then $G$ is $38$-Steinhaus
({\smc Kittrell \& Tsankov 09}, or {\smc Fremlin 03},
494O).   Find the smallest $k$ for which $G$ is
$k$-Steinhaus.   (\pound2.)

\medskip

{\bf GR$\dagger$.} For $k\ge 1$, let $G_k$ be the following 
$k$-double-move game
for two players.   I chooses $n\in\Bbb N$ and sets $X_0=[2n]^n$.   When I
has chosen $X_i$, where $i<k$, II chooses $S_i\subseteq[X_i]^2$ and I
chooses $X_{i+1}\subseteq X_i$ such that $[X_i]^2$ is either included in
$S_i$ or disjoint from $S_i$.   I wins if $\#(X_k)\ge 3$ and 
$\bigcap X_k=\emptyset$.   Which player has a winning strategy?  (\pound2.)
What if $k=1$?  (\pound1.)

\medskip

{\bf GS.}  Set $X=\BbbR^2$; let $\Cal C$ be the family of rectangles
$I\times J$ where $I$, $J\subseteq\Bbb R$ are connected sets,
and $\Cal E$ the algebra of subsets of $X$ generated by $\Cal C$.
Suppose that $\phi:\Cal C\to[0,1]$ is such that
$\sum_{C\in\Cal C_0}\phi C\le 1$ for every disjoint collection 
$\Cal C_0\subseteq\Cal C$.   Must there be an
additive functional $\lambda:\Cal E\to\coint{0,\infty}$ such that
$\lambda C\ge\phi C$ for every $C\in\Cal C$?  (\pound1.) 
Indeed, will there 
always be such a functional $\lambda:\Cal E\to[0,2]$?  (\pound1.)
(See {\smc Fremlin n11}, 2D.)

\medskip

{\bf GT.} Let $X$ be a compact Hausdorff space such that the space $P(X)$
of Radon probability measures on $X$ with its narrow topology (the
topology induced by the weak* topology of $C(X)^*$) is countably tight.
Does it follow that every member of $P(X)$ has countable Maharam type?
(If $\frak m>\omega_1$, yes.   See {\smc Plebanek \& Sobota p14}.)
(\pound2.)

\medskip

{\bf GU.} Let $\mu^{(2)}_{H1}$, $\mu^{(1)}_{H,1/2}$
be one-dimensional Hausdorff measure on $\Bbb R^2$ and
$\bover12$-dimensional Hausdorff measure on $\Bbb R$ respectively, for
their usual metrics.
Are the measure spaces $(\Bbb R^2,\mu^{(2)}_{H1})$ and
$(\Bbb R,\mu^{(1)}_{H,1/2})$ isomorphic?
%\pound

\bigskip

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\bigskip

\Centerline{\it In re mathematica ars proponendi quaestionem pluris
facienda est quam solvendi}
%this is from Ulam's version of Cantor's introduction to his thesis;
%"pluris" seems to be a form of "plus"

\bigskip

\noindent
Mathematics Department, University of Essex, Colchester CO3 3AT,
England;
{\tt david\penalty-100\@\penalty-100fremlin.org}.
This problem sheet is available at
{\tt https://www1.essex.ac.uk/maths/people/\penalty-100fremlin/problems.pdf}
and {\tt .ps}.

\end

Supplementary problems:

\medskip

(Kat\v etov) Is there, in ZFC, a normal Hausdorff space in which the
Baire and Borel $\sigma$-algebras coincide, but not all closed sets are
G$_{\delta}$?

\medskip

(P.J.Nyikos) In ZFC, is there a separable first-countable countably
compact Hausdorff space which is not compact?


	
